### Summary

The paper begins with a review of the essential points of*Lanczos's* orthogonalization procedure, which is of great importance for the determination of the eigenvalues of a real, but otherwise general matrix. Then several properties useful for numerical computation are proved:

If the degree of the reduced characteristic polynomial of the matrix is*m*, then it is possible to choose trial vectors*x, y* such that the iteration may be continued for exactly*m* steps. At that point the process must stop because the iterated vectors*x*_{m+1},*y*_{m+1} vanish (theorem 1).

If all the eigenvalues are real, then the co-diagonal elements*β*_{i} [see equation (12)] can be made arbitrarily small by proper choice of the trial vectors*x, y* (theorem 5), which considerably simplifies the evaluation of the eigenvalues and eigenvectors. Should, in addition, all the*β*_{i} still be positive (see theorem 4), then bounds for the eigenvalues may readily be given (theorem 3). Further remarks are made concerning matrices with complex eigenvalues.

Finally it is shown that by starting from a certain one-parameter family of trial vectors (11), the diagonal and co-diagonal elements*α*(*t*),*β*(*t*) are solutions of a system of differential equations (25).